Open Access
Article
On the Irregular Part of V -Statistics Multifractal Spectra for Systems with Non-Uniform Specification
ALEJANDRO MESÓNvericat@iflysib.unlp.edu.arInstituto De Física De Líquidos Y Sistemas Biológicos (IFLYSIB) Conicet–UNLP and Grupo De Aplicaciones Matemáticas Y EstadísticasDe La Facultad De Ingeniería (GAMEFI) UNLP s:La Plata, ArgentinaView full profile → , FERNANDO VERICATvericat@iflysib.unlp.edu.arInstituto De Física De Líquidos Y Sistemas Biológicos (IFLYSIB) Conicet–UNLP and Grupo De Aplicaciones Matemáticas Y EstadísticasDe La Facultad De Ingeniería (GAMEFI) UNLP s:La Plata, ArgentinaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 29 May 2014
- Accepted:
- 26 Dec 2014
- Published Online:
- 02 Jun 2015
- Article type:
- Article
- Language:
- EN
- Article no.:
- 1726037X.2015.1027108
- Pages:
- 1–26
Abstract
Let (X, f) be a dynamical system with X a compact metric space. Let Xundefined be the product of r-copies of X, r≥ 1, and Φ: Xundefined → R. The multifractal decomposition for V –statistics for Φ, f is defined as undefined.The set of points x ∊ X, for which the limit does not exist is called the irregular part, or historic set, of the spectrum. In this article we analyze the irregular part of the V -statistics for systems satisfying a weak form of the known Bowen specification property, called the non-uniform specification property. This concept was introduced by P. Varandas and allows to work in a nonuniformly hyperbolic context.
References
- V.Bergelson, Weakly mixing PET , Ergod. Th. and Dynam. Sys ., Vol. 7 , 337 – 349 ( 1987 ).
- J.Bourgain, Double recurrence and almost sure convergence , J. Reine Angew Math , Vol. 404 , 140 – 161 ( 1990 ).
- R.Bowen, Topological entropy for non-compact sets , Trans. Amer. Math. Soc ., Vol. 184 , 125 – 136 ( 1973 ).
- X.Zhou and E.Chen, Multifractal analysis of ergodic averages in some nonuniformly hyperbolic systems , preprint, arXiv:1310.2348v1 ( 2013 ).
- A. H.Fan, J.Schmeling and J.Wu, The multifractal spectrta of –statistics , preprint, arXiv:1206.3214v1 ( 2012 )
- A.Fan, D. J.Feng and J.Wu, Recurrence, dimension and entropy , J. London Math. Soc ., Vol. 64 , 229 – 244 ( 2001 ).
- A.Fan, I. M.Liao and J.Peyrière, Generic points in systems of specification and Banach valued Birkhoff averages , Disc. Cont. Dynam. Sys ., Vol. 21 , 1103 – 1128 ( 2008 ).
- H.Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szmerédi on arithmetic progressions , J. d′ Analyse Math ., Vol. 31 , 204 – 256 ( 1977 ).
- A.Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms , Inst. Hautes Studes Sci, Publ. Math , Vol. 51 , 137 – 173 ( 1980 ).
- A.Meson and F.Vericat, On The topological entropy of the irregular part of V-statitistics multifractal spectra , Journal of Dynamical Systems and Geometric Theories , Vol. 11 1 – 12 ( 2012 ).
- C. E.Pfister and W.G.Sullivan, On the topological entropy of saturated sets , Ergod. Th. and Dynam. Sys ., Vol. 27 , 1 – 29 ( 2007 ).
- F.Takens and E.Verbitski, On the variational principle for the topological entropy of certain non-compact sets , Ergod. Th. and Dynam. Sys ., Vol. 23 , 317 – 348 ( 2003 ).
- D.Thompson, The irregular set for maps with the specification property has full topological pressure , Dynam Sys: An International Journal , Vol. 25 25 – 51 ( 2010 ).
- P.Varandas, Non-uniform specification and large deviations for weaks Gibbs measures , J. Stat. Phys ., Vol. 146 , 330 – 358 ( 2012 ).
- Z.Yin. E.Chen and X.Zhou, The historic set of ergodic averages in some non-uniformly hyperbolic systems , preprint, arXiv:1405.3339v1 ( 2014 )
- L. S.Young, Large deviations in dynamical systems , Trans. Amer. Math. Soc ., Vol. 318 , 525 – 543 ( 1990 ).
- P.Walters, An introduction to Ergodic Theory , Springer-Verlag , 1982 .
Views: 39Downloads: 91Citations: 1
